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On the prime radical of a module over a commutative ring. (English) Zbl 0776.13003

All rings will be commutative with identity and all modules will be unitary. The prime radical of a module \(M\) over a ring \(R\) is the intersection of \(M\) with all prime submodules of \(M\). We shall denote by \(W(M)\) the set of elements of \(M\) which may be written in the form \(r_ 1m_ 1+\cdots+r_ nm_ n\), where \(n\) is a positive integer, \(r_ i \in R\), \(m_ i \in M\), and \(r^ k_ im_ i=0\;(1 \leq i \leq N)\), for some positive integer \(k\). Among other results the authors prove the following lemma:
Let \(R\) be a Noetherian domain of dimension 1 and let \(M\) be any \(R\)- module. Then \(\text{rad} M=\bigcup \text{rad} L\), where the union is taken over all finitely generated submodules \(L\) of \(M\).
Corollary. Let \(R\) be any ring and \(M\) any projective \(R\)-module. Then \(\text{rad} M=W(M)\).
Theorem. Let \(R\) be a Dedekind domain and \(M\) any \(R\)-module. Then \(\text{rad} M=W(M)\).
An \(R\)-module \(M\) “satisfies the radical formula” if \(\text{rad} (M/N)=W(M/N)\) for any submodule \(N\) of \(M\). If every \(R\)-module satisfies the radical formula we say that \(R\) does (STRF). The authors prove that for any \(R\)-module \(M\), \(M\) STRF is equivalent to every semiprime submodule of \(M\) being an intersection of prime submodules of \(M\) and \(W[(M/W(M)]=0\). – An example of a semiprime submodule which is not the intersection of prime submodules is given to answer a query of Dauns, and several equivalences to STRF are given for a Noetherian UFD.

MSC:

13A10 Radical theory on commutative rings (MSC2000)
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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[1] Dauns, J. 1980. Prime modules and one–sided ideals, in ”Ring Theory and Algebra 111”. (Proceedings of the Third Oklahoma Conference), 1980, New York. Dekker. B. R. McDonald (editor) 301 – 344.
[2] DOI: 10.1090/S0002-9947-1952-0046349-0
[3] Matsumura H. Commutative algebra Second edition Benjamin/Cummings Reading 1980
[4] DOI: 10.4153/CMB-1986-006-7 · Zbl 0546.13001
[5] DOI: 10.1080/00927879108824205 · Zbl 0745.13001
[6] McCasland R. L. Smith P. F. Prime submodules of Noetherian modules preprint. · Zbl 0814.16017
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